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On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains

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  • Martin Costabel

Abstract

We construct a bounded C1 domain Ω in Rn for which the H3/2 regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in C∞(Ω¯) such that the solution of Δu=f in Ω and either u=0 on ∂Ω or ∂nu=0 on ∂Ω is contained in H3/2(Ω) but not in H3/2+ε(Ω) for any ε>0. An analogous result holds for Lp Sobolev spaces with p∈(1,∞).

Suggested Citation

  • Martin Costabel, 2019. "On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains," Mathematische Nachrichten, Wiley Blackwell, vol. 292(10), pages 2165-2173, October.
    • Handle: RePEc:bla:mathna:v:292:y:2019:i:10:p:2165-2173
    DOI: 10.1002/mana.201800077
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    Cited by:

    1. Cornelia Schneider & Flóra Orsolya Szemenyei, 2023. "Besov regularity of inhomogeneous parabolic PDEs," Partial Differential Equations and Applications, Springer, vol. 4(5), pages 1-61, October.

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